Solution Manual for Experimental Design: Procedures for the Behavioral Sciences, 4th Edition

CHAPTER 2 Experimental Designs: An Oveview 2. d. (i) RB-3 design (ii) H0: µ.1 = µ.2 = µ.3 (iii) Yij = µ + αj + πi + εij (i = 1, . . . , 14; j = 1, . . . , 3) e. (i) t test for dependent samples (ii) H0: µ.1 ≤ µ.2, where µ.1 and µ.2 denote the population means for English-Canadian and French-Canadian students, respectively. (iii) Yij = µ + αj + πi + εij (i = 1, . . . , 50; j = 1, 2) f. (i) CRF-62 design (ii) H0: µ1. = µ2. = . . . = µ6.; H0: µ.1 = µ.2 ; H0: µ jk – µ jk ! – µ j !k + µ j !k ! = 0 for all j and k (iii) Yijk = µ + αj + βk + (αβ)jk + εi(jk) (i = 1, . . . , 50; j = 1, . . . , 6; k = 1, 2) g. (i) CR-3 design (ii) H0: µ1 = µ2 = µ3 (iii) Yij = µ + αj + εi(j) (i = 1, . . . , 30; j = 1, . . . , 3) 3. a. The grand mean is the average value around which the treatment means vary. b. A treatment effect is the deviation of the grand mean from a treatment mean. c. An error effect is all effects not attributable to a treatment level or treatment combination. 4. b. A completely randomized design is the simplest design to lay out and analyze. The randomization procedures for the randomized block and Latin square designs are more complex than those for the completely randomized design, but the latter designs enable a researcher to isolate the effects of one nuisance variable or, in the case of the Latin square design, two nuisance variables. 5. c. a1b1, a1b2, a1b3, a2b1, a2b2, a2b3, a3b1, a3b2, a3b3 d. a1b1, a1b2, a2b1, a2b2, a3b1, a3b2, a4b1, a4b2 e. a1b1c1, a1b1c2, a1b2c1, a1b2c2, a2b1c1, a2b1c2, a2b2c1, a2b2c2, a3b1c1, a3b1c2, a3b2c1, a3b2c2 3 Chapter 2 Experimental Designs: An Overview 6. d. CR-5 design with n = 6 Group1 ! ” # Treat. Level Dep. Var. Subject1 a1 Y11 ! ! ! Subject6 a1 Y61 Y.1 Group2 ! ” # Subject1 ! Subject6 a2 ! a2 Y12 ! Y62 Y.2 Group3 ! ” # Subject1 ! a3 ! Subject6 a3 Y13 ! Y63 Y.3 Group4 ! ” # Subject1 ! a4 ! Subject6 a4 Y14 ! Y64 Y.4 Group5 ! ” # Subject1 ! a5 ! Subject6 a5 Y15 ! Y65 Y.5 e. t test for dependent samples with n = 7 Treat. Level Dep. Var. Treat. Level Dep. Var. Block1 Block2 Block3 a1 a1 a1 Y11 Y21 Y31 a2 a2 a2 Y12 Y22 Y32 ! ! ! ! Block7 a1 Y71 a2 Y.1 ! Y7 2 Y.2 4 Experimental Designs: An Overview Chapter 2 f. RB-4 design with n = 6 Treat. Dep. Level Var. Treat. Level Dep. Var. Treat. Level Dep. Var. Treat. Dep. Level Var. Block1 a1 Y11 a2 Y12 a3 Y13 a4 Y14 Y1. Block2 a1 Y21 a2 Y22 a3 Y23 a4 Y24 Y2. Block3 a1 Y31 a2 Y32 a3 Y33 a4 Y34 Y3. ! ! ! ! ! ! ! ! ! ! Block6 a1 Y61 a2 Y62 a3 Y63 a4 Y64 Y6. Y.1 Y.2 Y.3 g. CRF-222 design with n = 3 Subject1 Subject2 Subject3 ! # Group1 ” # $ Treat. Comb. Dep. Var. a1b1c1 a1b1c1 a1b1c1 Y1111 Y2111 Y3111 Y.111 Group2 ! # ” # $ Subject1 Subject2 Subject3 a1b1c2 a1b1c2 a1b1c2 Y1112 Y2112 Y3112 Y.112 Group3 ! # ” # $ Subject1 Subject2 Subject3 a1b2c1 a1b2c1 a1b2c1 Y1121 Y2121 Y3121 Y.121 Group8 ! # ” # $ ! ! ! Subject1 Subject2 Subject3 a2b2c2 a2b2c2 a2b2c2 Y1222 Y2222 Y3222 Y.222 5 Y.4 Chapter 2 h. Experimental Designs: An Overview LS-3 design with n = 3 Group1 ! # ” # $ Subject1 Subject2 Subject3 Treat. Comb. Dep. Var. a1b1c1 a1b1c1 a1b1c1 Y1111 Y2111 Y3111 Y.111 Group2 ! # ” # $ Subject1 Subject2 Subject3 a1b2c3 a1b2c3 a1b2c3 Y1123 Y2123 Y3123 Y.123 Group3 ! # ” # $ Subject1 Subject2 Subject3 a1b3c2 a1b3c2 a1b3c2 Y1132 Y2132 Y3132 Y.132 Group4 ! # ” # $ Subject1 Subject2 Subject3 a2b1c2 a2b1c2 a2b1c2 Y1212 Y2212 Y3212 Y.212 Group9 ! # ” # $ ! ! ! Subject1 Subject2 Subject3 a3b3c1 a3b3c1 a3b3c1 Y1331 Y2331 Y3331 Y.331 6 Chapter 2 Experimental Designs: An Overview Treat. Level Dep. Var. Treat. Level Dep. Var. 8. d. RB-3 design with n = 14 Treat. Dep. Level Var. Block1 a1 Y11 a2 Y12 a3 Y13 Y1. Block2 a1 Y21 a2 Y22 a3 Y23 Y2. Block3 a1 Y31 a2 Y32 a3 Y33 Y3. ! ! ! ! ! ! ! ! a1 Y14, 1 a2 Y14, 2 a3 Y14, 3 Y14. Block14 Y.1 Y.2 Y.3 e. t test for dependent samples with n1 and n2 = 50 Treat. Level Dep. Var. Treat. Level Dep. Var. Block1 Block2 Block3 a1 a1 a1 Y11 Y21 Y31 a2 a2 a2 Y12 Y22 Y32 ! ! ! ! Block50 a1 Y50, 1 a2 Y.1 ! Y50, 2 Y.2 7 Chapter 2 Experimental Designs: An Overview f. CRF-62 design with n = 50 Group1 ! # ” # $ Treat. Comb. Dep. Var. Subject1 a1b1 Y111 ! ! Subject50 a1b1 ! Y50, 11 Y.11 Group2 ! # ” # $ Subject1 a1b2 ! ! Subject50 a1b2 Y112 ! Y50, 12 Y.12 Group3 ! # ” # $ Subject1 a2b1 ! ! Subject50 a2b1 Y121 ! Y50, 21 Y.21 Group4 ! # ” # $ Subject1 a2b2 Y122 ! ! ! Subject50 a2b2 Y50, 22 Y.22 ! Group12 #” # $ ! ! ! Subject1 a6b2 Y162 ! ! Subject50 a6b2 ! Y50, 62 Y.62 8 Experimental Designs: An Overview Chapter 2 g. CR-3 design with n = 30 Group1 Treat. Level Dep. Var. a1 Y11 ! ! #! Animal1 ” ! #$ Animal30 a1 Y30, 1 Y.1 Group2 !# Animal1 ” ! #$ Animal30 a2 ! a2 Y12 ! Y30, 2 Y.2 Group3 #! Animal1 ” ! #$ Animal30 a3 Y13 ! ! a3 Y30, 3 Y.3 a1 a2 9 8 a3 7 6 5 4 b1 b2 b3 10 hrs. 15 hrs. 20 hrs. Hours of deprivation Running time (sec.) Running time (sec.) 9. a. b1 9 b2 8 b3 7 6 5 4 a1 a2 a3 Small Medium Large Magnitude of reinforcement b. As the number of hours of deprivation increases, the difference in running time among the three reinforcement conditions decreases. 12. a. A scientific hypothesis is a testable supposition that is tentatively adopted to account for certain facts and to guide in the investigation of others. A statistical hypothesis is a statement about one or more parameters of a population or the functional form of a population. b. (i) Alternative hypothesis (ii) Null hypothesis 15. a. State the null and alternative hypotheses—H0: µ = 45, H1: µ ≠ 45. Specify the test statistic—t = (Y ! µ 0 ) / (“ˆ / n ) . Specify the sample size—n = 27, and the sampling distribution—t distribution. Specify the level of significance—α = .05. Obtain random 9 Chapter 2 Experimental Designs: An Overview samples of size n = 27, compute t, and make a decision. b. Reject the null hypothesis if t falls in either the lower or upper 2.5% of the sampling distribution of t; otherwise, do not reject the null hypothesis. If the null hypothesis is rejected, conclude that the mean for children in the experimental program is not equal to the mean for ninth-graders who have been observed during the past several years; if the null hypothesis is not rejected, do not draw this conclusion. c. Critical region ! = .025 f (t) Critical region ! = .025 Reject H0 Reject H0 t d. t = Y ! µ0 Don’t reject H0 = 52.5 ! 45.0 = 7.5 = 2.60 , p = .015. 2.89 “ˆ / n 15 / 27 The population mean for children in the experimental program was not equal to the mean for ninth-graders who have been observed during the past several years. The difference between children who did or did not participate in the experimental program, 52.5 versus 45.0, was statistically significant, t(26) = 2.60, p = .015. e. d = 52.5 ! 45 / 15 = 0.5 ; this is a medium size effect. Y! f. 52.5 ! t.05/2, 26″ˆ <µ <Y + n 2.056(15) 27 t.05/2, 26"ˆ n < µ < 52.5 + 2.056(15) 27 46.6 < µ .05 may preclude the publication of the research. Refining the experimental methodology so as to decrease the size of the population standard deviation may be prohibitively expensive. Increasing the magnitude of the treatment effects considered worth detecting may not be appropriate. 18. a. State the null and alternative hypotheses—H0: µ1 – µ2 ≤ 0, H1: µ1 – µ2 > 0. Specify the #1 1& test statistic— t = (Y.1 ! Y.2 ) / “ˆ 2Pooled % + ( . Specify the sample size—n1 = 24, n2 $ n1 n2 ‘ = 23, and the sampling distribution—t distribution. Specify the level of significance—α = .05. Randomly assign N = 47 subjects to the two game types, compute t, and make a decision. b. Reject the null hypothesis if t falls in the upper 5% of the sampling distribution of t; otherwise, do not reject the null hypothesis. If the null hypothesis is rejected, conclude that the risk-related cognitions of men who play racing video games is higher than that for the men who play the neutral games; if the null hypothesis is not rejected, do not draw this conclusion. 11 Chapter 2 Experimental Designs: An Overview c. Critical region ! = .05 f(t) t Reject H0 d. Don’t reject H0 ( n1 ” 1)!12 + ( n2 ” 1)! 22 (24 ” 1)(1.3)2 + (23″ 1)(1.2)2 2 ! Pooled = = = 1.568 ( n1 ” 1) + ( n2 ” 1) (24 ” 1) + (23″ 1) #1 1& ” 1 1% t = (Y.1 ! Y.2 ) / “ˆ 2Pooled % + ( = (7.54 ! 6.41) / 1.568 $ + ‘ # 24 23 & $ n1 n2 ‘ = 1.13 / 0.365 = 3.09 . The p value is less than .002. The mean risk-related cognitions for men who played the racing video games was higher than that for the men who played the neutral games. The difference between the means, 7.54 versus 6.41, was statistically significant, t(45) = 3.09, p < .002. e. g = Y.1 ! Y.2 / "ˆ Pooled = 7.54 ! 6.41 / 1.25 = 0.90 ; this is a large effect. #1 1& (Y.1 ! Y.2 ) ! t.05,45 "ˆ 2Pooled % + ( < µ1 ! µ 2 $ n1 n2 ' f. " 1 1% (7.54 ! 6.41) ! 1.679 1.568 $ + ' < µ1 ! µ 2 # 24 23 & 0.52 0 Type I error Correct rejection α = .05 1 – !ˆ = 1 – .15 = .85 19. c. p < .0005 d. p < .1048 20. c. t = 2.402 d. t = 3.601 13